蓋根鮑爾多項式 蓋根鮑爾多項式
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{\displaystyle C_{n}^{(\alpha )}}
超球多項式 ，是定義在區間
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{\displaystyle [-1,1]}
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{\displaystyle (1-x^{2})^{\alpha -1/2}}
正交多項式 。它是勒讓德多項式 和切比雪夫多項式 的推廣，又是雅可比多項式 的特殊情況。它以奧地利數學家Leopold Gegenbauer 命名。
蓋根鮑爾多項式具有若干性質：
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{\displaystyle {\frac {1}{(1-2xt+t^{2})^{\alpha }}}=\sum _{n=0}^{\infty }C_{n}^{(\alpha )}(x)t^{n}.}
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{\displaystyle {\begin{aligned}C_{0}^{\alpha }(x)&=1\\C_{1}^{\alpha }(x)&=2\alpha x\\C_{n}^{\alpha }(x)&={\frac {1}{n}}[2x(n+\alpha -1)C_{n-1}^{\alpha }(x)-(n+2\alpha -2)C_{n-2}^{\alpha }(x)].\end{aligned}}}
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{\displaystyle (1-x^{2})y''-(2\alpha +1)xy'+n(n+2\alpha )y=0.\,}
當 α = 1/2, 方程約化為勒讓德方程, 蓋根鮑爾多項式約化為勒讓德多項式 .
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{\displaystyle C_{n}^{(\alpha )}(z)={\frac {(2\alpha )_{n}}{n!}}\,_{2}F_{1}\left(-n,2\alpha +n;\alpha +{\frac {1}{2}};{\frac {1-z}{2}}\right).}
(Abramowitz & Stegun p. 561 （頁面存檔備份 ，存於網際網路檔案館 ）). 其中(2α)n 上升階乘冪 . 具體來說,
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{\displaystyle C_{n}^{(\alpha )}(z)=\sum _{k=0}^{\lfloor n/2\rfloor }(-1)^{k}{\frac {\Gamma (n-k+\alpha )}{\Gamma (\alpha )k!(n-2k)!}}(2z)^{n-2k}.}
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{\displaystyle C_{n}^{(\alpha )}(x)={\frac {(2\alpha )_{n}}{(\alpha +{\frac {1}{2}})_{n}}}P_{n}^{(\alpha -1/2,\alpha -1/2)}(x).}
因而滿足羅德里格公式
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{\displaystyle C_{n}^{(\alpha )}(x)={\frac {(-2)^{n}}{n!}}{\frac {\Gamma (n+\alpha )\Gamma (n+2\alpha )}{\Gamma (\alpha )\Gamma (2n+2\alpha )}}(1-x^{2})^{-\alpha +1/2}{\frac {d^{n}}{dx^{n}}}\left[(1-x^{2})^{n+\alpha -1/2}\right].}
當n ≠ m 時，對於固定的α 和權函數
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{\displaystyle w(z)=\left(1-z^{2}\right)^{\alpha -{\frac {1}{2}}}.}
蓋根鮑爾多項式在區間[−1, 1]上加權正交 (Abramowitz & Stegun p. 774 （頁面存檔備份 ，存於網際網路檔案館 ）)
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{\displaystyle \int _{-1}^{1}C_{n}^{(\alpha )}(x)C_{m}^{(\alpha )}(x)(1-x^{2})^{\alpha -{\frac {1}{2}}}\,dx=0.}
歸一性：
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{\displaystyle \int _{-1}^{1}\left[C_{n}^{(\alpha )}(x)\right]^{2}(1-x^{2})^{\alpha -{\frac {1}{2}}}\,dx={\frac {\pi 2^{1-2\alpha }\Gamma (n+2\alpha )}{n!(n+\alpha )[\Gamma (\alpha )]^{2}}}.}
蓋根鮑爾多項式作為勒讓德多項式的擴展經常出現在勢理論 和譜分析 中. R n 牛頓勢 可以在α = (n − 2)/2情況下展開為蓋根鮑爾多項式,
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{\displaystyle {\frac {1}{|\mathbf {x} -\mathbf {y} |^{n-2}}}=\sum _{k=0}^{\infty }{\frac {|\mathbf {x} |^{k}}{|\mathbf {y} |^{k+n-2}}}C_{n,k}^{(\alpha )}(\mathbf {x} \cdot \mathbf {y} ).}
當n = 3, 可以得到引力勢 的勒讓德展開。類似的表達式還有球中泊松核 的展開(Stein & Weiss 1971 ).
當只考慮x 時，
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{\displaystyle C_{n,k}^{((n-2)/2)}(\mathbf {x} \cdot \mathbf {y} )}
球諧函數 。
蓋根鮑爾多項式在正定函數 理論中亦有涉及。
Abramowitz, Milton; Stegun, Irene Ann (編). Chapter 22 . Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series 55 Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first. Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. 1983: 773. ISBN 978-0-486-61272-0LCCN 64-60036 MR 0167642 Bayin, S.S., Mathematical Methods in Science and Engineering, Wiley, 2006 Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F., Orthogonal Polynomials , Olver, Frank W. J. ; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (編), NIST Handbook of Mathematical Functions , Cambridge University Press, 2010, ISBN 978-0521192255MR 2723248 Stein, Elias ; Weiss, Guido , Introduction to Fourier Analysis on Euclidean Spaces, Princeton, N.J.: Princeton University Press, 1971, ISBN 978-0-691-08078-9 Suetin, P.K., Ultraspherical polynomials , Hazewinkel, Michiel (編), 数学百科全书 , Springer , 2001, ISBN 978-1-55608-010-4 
![{\displaystyle [-1,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51e3b7f14a6f70e614728c583409a0b9a8b9de01)
![{\displaystyle {\begin{aligned}C_{0}^{\alpha }(x)&=1\\C_{1}^{\alpha }(x)&=2\alpha x\\C_{n}^{\alpha }(x)&={\frac {1}{n}}[2x(n+\alpha -1)C_{n-1}^{\alpha }(x)-(n+2\alpha -2)C_{n-2}^{\alpha }(x)].\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7f7e040cb401c72d41eab0bda662f115229feed)
![{\displaystyle C_{n}^{(\alpha )}(x)={\frac {(-2)^{n}}{n!}}{\frac {\Gamma (n+\alpha )\Gamma (n+2\alpha )}{\Gamma (\alpha )\Gamma (2n+2\alpha )}}(1-x^{2})^{-\alpha +1/2}{\frac {d^{n}}{dx^{n}}}\left[(1-x^{2})^{n+\alpha -1/2}\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07eff10ed90eb66bf06c2b329b1d266114dff433)
![{\displaystyle \int _{-1}^{1}\left[C_{n}^{(\alpha )}(x)\right]^{2}(1-x^{2})^{\alpha -{\frac {1}{2}}}\,dx={\frac {\pi 2^{1-2\alpha }\Gamma (n+2\alpha )}{n!(n+\alpha )[\Gamma (\alpha )]^{2}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9b5696990f291fed55949f3b9e2f1669b9f4c83)
